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Abstract

Accidental degeneracy in a photonic crystal consisting of a square array of elliptical dielectric cylinders leads to both a semi-Dirac point at the center of the Brillouin zone and an electromagnetic topological transition (ETT). A perturbation method is deduced to affirm the peculiar linear-parabolic dispersion near the semi-Dirac point. An effective medium theory is developed to explain the simultaneous semi-Dirac point and ETT and to show that the photonic crystal is either a zero-refractive-index material or an epsilon-near-zero material at the semi-Dirac point. Drastic changes in the wave manipulation properties at the semi-Dirac point, resulting from ETT, are described.

Figures (5)

(a) The band structure of the 2D PhC composed of a square array of elliptical dielectric cylinders. The inset shows the unit cell of the PhC. A doubly-degenerate state in the center of the Brillouin zone is found near the dimensionless frequency, 0.540, marked as “A”. In the vicinity of this point, the dispersion relation is linear along the ΓXdirection and quadratic along theΓY direction, which is shown more clearly in Fig. 3(d). Near point “A”, there is another state in the center of the Brillouin zone, marked as “B”. The states at points “A” and “B” are used in the perturbation theory. The branches highlighted by black and blue dots are used to compute the effective medium parameters, which are shown in Fig. 4(a). (b) and (c) Enlarged views of the band structure for smaller and larger elliptical cylinders. The doubly-degenerate state shown in (a) splits into two single states, marked as A1 and A2, where A1 corresponds to a dipolar state and A2 corresponds to a monopolar state.

(a) and (b) The three-dimensional band structure of the PhC. The upper surface is a semi-Dirac cone. Near its bottom, it is linear in Δk along all directions except for the ΓY direction, which is quadratic. It touches the lower surface at the Brillouin zone center near the dimensionless frequency, 0.54. The lower surface is flat in one direction and bends down along the other directions. (c) and (d) The iso-frequency surfaces of the lower and higher branches, where hyperbolic and elliptical surfaces are found, respectively.

(a) The electric field pattern of the eigenstate marked as “B” in Fig. 1(a). Dark red and dark blue indicate the maximum positive and negative values, respectively. This is a dipolar state with a magnetic field parallel to the x-axis, indicated by the arrows. (b) and (c) The electric field patterns of the doubly-degenerate states marked as “A” in Fig. 1(a). A monopolar and a dipolar state with the magnetic field (arrows) perpendicular to the x-axis are evident. (d) An enlarged view of the band structure near the doubly-degenerate point. The dots are calculated by COMSOL. Linear dispersion is seen along the ΓX direction, while a quadratic dispersion relation is manifest along the ΓY direction. Red solid lines and green solid curves are obtained from the perturbation theory. The blue dashed curves represent the results of quadratic fitting. (e) The same as (d) but along the ΓMdirection. A linear dispersion relation is seen again.

(a) Effective medium parameters evaluated with a boundary effective medium theory using the eigenstates highlighted by solid dots in Fig. 1(a). The blue triangles and black squares represent the effective permittivity εeff calculated by using the eigenstates along the ΓYand ΓX directions, respectively. They almost overlap, indicating that εeffis a scalar and does not depend on the direction. The red circles represent μyeff, which crosses zero simultaneously with εeffat the semi-Dirac point. The green triangles represent μxeff, which crosses zero at dimensionless frequency 0.487. Note that both the blue and green triangles are missing from the frequency regime at 0.487 to 0.540, which corresponds to a band gap along the ΓY direction. No eigenstates are thus available to evaluate the related effective medium parameters. (b)-(d) The electric field for a plane wave impinging on a PhC slab in a waveguide whose walls have perfect magnetic conductor boundary conditions at the semi-Dirac frequency 0.540. Dark red and dark blue indicate the maximum positive and negative values, respectively. (b) The real part of the electric field when the incident wave is along the ΓY direction. The transmitted field is very weak. The imaginary part is orders of magnitude smaller than the real part, which is why it is not shown here. (c) and (d) The real and imaginary parts of the electric field when the incident wave is along the ΓXdirection. Both suggest that there is no phase change in the sample, which is a typical property of a ZIM.

A point source is placed inside the center of a square sample of 16-by-16 rods. (a) and (c) show the electric field patterns when the source frequency is below (0.520) and slightly above (0.544) the semi-Dirac point, respectively. Beam splitting and directional beam shaping are observed. (b) The radial flux as a function of the angle for the case simulated in (a). (d) The same as (c) but the sample is replaced by its effective medium. A similar pattern to that shown in (c) is found. Dark red and dark blue indicate the maximum positive and negative values, respectively.